Introduction to Time Series Analysis of Macroeconomic- and Financial-Data. Lecture 2: Testing & Dependence over time

Transcription

1 Introduction Introduction to Time Series Analysis of Macroeconomic- and Financial-Data Felix Pretis Programme for Economic Modelling Oxford Martin School, University of Oxford Lecture 2: Testing & Dependence over time Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

2 Summary from Yesterday Yesterday: Today: Econometric Models Ordinary Least-Squares (OLS) Regression Line of best fit Hypothesis testing Goodness of fit More than one explanatory variable Is the model well-specified? Time-series: dependence over time! Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

3 Including more variables Extend model to include two explanatory variables: X t and Z t. Triplets of data, (Y t, X t, Z t ): Three economic variables occurring simultaneously. Assumptions: (i) (Y t, X t, Z t ) independent across t. (ii) Identical conditional distribution: (Y t X t, Z t ) (β 1 + β 2 X t + β 3 Z t, σ 2 ). (iii) X t and Z t exogenously determined for Y t. (iv) A parameter space exists: β 1, β 2, β 3, σ 2 R R +, Gives model: Y t = β 1 + β 2 X t + β 3 Z t + ɛ t, ɛ t N0, σ 2 (1) Parameter interpretation as before except ceteris paribus: EY t X t, Z t X t = β 2, (2) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

4 Baseball Attendance Red Sox Game Attendance Winning Percentage (PCT) US Populations - to scale Attendance (att pop) US Unemployment Rate Winning Percentage PCT Attendance Attendance e US Population us_pop US Unemployment Rate US_unempl_rate 2.0e e Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

5 Baseball Attendance: Model Model: %Attendance t = β 1 + β 2 PCT t + β 3 Unemp. t + ɛ t Attendance/Population att_pop att_pop PCT att_pop US_unempl_rate Attendance/Population Attendance/Population Winning Percentage US Unemployment Rate Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

6 %Attendance Fitted Residual (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

7 Baseball Attendance: Model Output EQ( 1) Modelling att_pop by OLS The dataset is: red_sox_1948.in7 The estimation sample is: Coefficient Std.Error t-value t-prob Part.Rˆ2 Constant PCT US_unempl_rate sigma RSS Rˆ F(2,49) = [0.000]** Adj.Rˆ log-likelihood no. of observations 52 no. of parameters 3 mean(att_pop) se(att_pop) When the log-likelihood constant is NOT included: AIC SC HQ FPE e-006 When the log-likelihood constant is included: AIC SC HQ FPE e-005 AR 1-2 test: F(2,47) = [0.0010]** ARCH 1-1 test: F(1,50) = [0.2773] Normality test: Chiˆ2(2) = [0.9400] Hetero test: F(4,47) = [0.1969] Hetero-X test: F(5,46) = [0.2983] RESET23 test: F(2,47) = [0.5127] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

8 t-testing Procedure to Check Variable Significance Create null hypothesis: H0 : β 2 = 0. Variable is insignificant: Regression coefficient has zero mean. Divide estimator (ˆβ 2 here) by estimated standard error ( V[ˆβ 2 ] here). This calculates the t-statistic. Check t-statistic against critical values from t distribution (t T here): If 5% significance level then check against tables or take roughly ±2. 1 If t-statistic greater than ±2 then we reject null hypothesis. Otherwise we do not reject null hypothesis. 1 Recall test is two-tailed: We do not care whether significance is positive or negative! Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

9 p-values In regression output, the t-prob column contains p-values. In output of other tests (details later), p-values are in [square brackets]. p-value is probability of rejecting null-hypothesis if null hypothesis is true probability of observing test-statistic under the null hypothesis if sufficiently unlikely (small), reject H 0 Calculated by imposing null hypothesis, using data and assumed distribution. If p-value very small, null hypothesis is statistically unlikely: If p-value below significance level α (e.g. 0.05) we reject null. p-value probability of falsely rejecting null hypothesis. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

10 Testing Joint Hypotheses Baseball Model: Att pop t = ( ) ( ) PCT t ( ) Unemp t In multivariate regression, F-test of joint significance different from t-statistics. t-testing is for individual regression coefficients: Testing the (null) hypothesis that each one individually is insignificant. F-testing is for combinations of regression coefficients: Testing the (null) hypothesis that together they are insignificant. Unemployment individually insignficant, variables jointly significant? F(2,49) = [0.000]** Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

11 We test joint hypotheses using the F-test: Calculate test statistic and find p-value: Probability of observing test statistic under null We find test statistic and distribution: F = (RSS R RSS U )/(K U K R ) RSS U /(T K U ) F KU K R,T K U. (3) Baseball model: H 0 : β 2 = 0.025, β 3 = 0. We could impose null by creating Y t X t variable but much easier: Use PcGive: Test then General Restrictions. Each variable denoted by ampersand (&) and number (See key beneath). Each restriction is line of code; must be ended with semi-colon ;. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

12 F-Distribution Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

13 F-Testing in general Common use of F test: Overall significance of model. More relevant when we add even more regressors: Thursday and Friday. F-test statistic: F = (RSS R RSS U )/(K U K R ) RSS U /(T K U ) = (TSS RSS)/(K 1). RSS/(T K) (4) ESS = TSS RSS: How much do variables explain of variation? Above constant: Base model is still one-variable model Y t = β 1 + ɛ t. Form of F yields distribution: F K 1,T K and hence critical values. Null hypothesis: Explanatory variables are jointly statistically insignificant. E.g. H 0 : β 2 = β 3 = 0. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

14 Diagnostics: Testing Model Specification Our assumptions imply ɛ t has independent (i) identical (ii) distribution. Often written as ɛ t iid(0, σ 2 ) (distribution need not be Normal). Y t = β 1 + β 2 X t + β 3 Z }{{} t + ɛ t, ɛ t iid(0, σ 2 ). }{{} Our model: What we know Errors: What we don t know (5) Important because want errors (mistakes) from our model to be: Unrelated to each other. Always of a similar size. This is central principle of all economic and econometric modelling. If not then our model does not include some important information! More importantly, ˆβ 2, ˆβ 3 may be biased and inefficient. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

15 Baseball Attendance: Specification Residuals (scaled) Density Residuals N(s= ) 350 Random vs. Patterns? Normal? AR 1-2 test: F(2,47) = [0.0010]** ARCH 1-1 test: F(1,50) = [0.2773] Normality test: Chiˆ2(2) = [0.9400] Hetero test: F(4,47) = [0.1969] Hetero-X test: F(5,46) = [0.2983] RESET23 test: F(2,47) = [0.5127] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

16 Estimated regression model: Y t = ˆβ 1 + ˆβ 2 X t + ˆβ 3 Z t + ˆɛ t. (6) Everything so far depends on iid assumption holding: Unbiasedness and efficiency of estimators. Accuracy of standard errors and all other test statistics. Must test whether assumptions hold: What do ˆɛ t look like? Tests called misspecification tests or diagnostic tests. Model: Assume ɛ t iid(0, σ 2 ) X 1,t,..., X K,t. Check Functional form. Data transformations. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

17 What can go wrong? 1 Identical distribution: Is ɛ t (0, σ 2 )? Mean must be zero (if constant included), but variance can change. Heteroskedasticity: Vɛ t = σ 2 t. 2 Independent distribution: Is Corr [ɛ s, ɛ t ] = E [ɛ s ɛ t ] = 0, for all s t? If not, autocorrelation. Variance not autocorrelated (ARCH) 3 Normal distribution: Is ɛ t N0, σ 2? Residuals may have very different distribution. 4 Functional Form: Model specification test = will consider in turn! Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

18 Testing in OxMetrics OxMetrics has very developed ways (tests) for detecting misspecification. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

19 When conducting testing in OxMetrics, will come across many different test. t-test: Tests on individual regression coefficients. F-test: Tests on combinations of regression coefficients. Additionally three types of test based on Likelihood: Maximum likelihood is alternative, more flexible form of estimation. Choose parameters (β 1, β 2 ) to maximise likelihood of having observed our data. Likelihood function is probability density as function of parameters. Tests are based on different aspects of likelihood function: Likelihood Ratio (LR) test: How likely is null hypothesis? Wald test: Computes only under alternative hypothesis. Lagrange Multipler (LM) test: Evaluates only under null hypothesis. Type of test not too important; p-values are the fundamental concept. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

20 Heteroskedasticity Assumption of ɛ t (0, σ 2 ) violated. Variance changes through sample: ɛ t (0, σ 2 t). Examples: Stock market more volatile in financial crisis. Inflation more volatile when higher. Loss of precision: Efficiency of estimator relies on constant σ 2 (homoskedasticity). Standard errors larger than they should be. Possible mistake: May conclude variable insignificant that is actually significant. On average we still get right answer, but more possibility we make mistakes. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

21 Heteroskedasticity: Detection Detection: Use residuals ˆɛ t in place of errors ɛ t. Graphic inspection: Does the variance look constant? Residuals (scaled) are standard graphics output in OxMetrics Example Heteroskedastic Errors 2.5 Residuals Residuals (scaled) Baseball Attendance Model Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

22 Formal Detection White test of heteroskedasticity: regress ˆɛ 2 t on X 1,t,..., X K,t, and X 2 1,t,..., X2 K,t : Does residual variance depend on regressors included in model, or their squares? Square of variable rough approximation to variance. Model: ˆɛ 2 t = α 0 + α 1 X 1,t + + α K X K,t + α K+1 X 2 1,t + + α 2K X 2 K,t + v t. Null hypothesis: H0 : α 1 = = α 2K = 0, (7) Absence of heteroskedasticity, or homoskedasticity (constant variance). F-test of joint significance: F Het = R 2 Het /m ( 1 R 2 Het ) / (T m) F m,t m. (8) Hetero test: F(4,643) = [0.0000]** Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

23 F-Distribution Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

24 Heteroskedasticity: Causes Formal Test in Attendance Model: Hetero test : F(4, 47) = [0.1969] Causes: Omitted Variables: Systematic variation in residuals. Unincluded variable may explain change in variation. E.g. Age may affect how volatile income is across individuals. Wrong functional form/data transformation: Logarithmic transformation can stabilise variance in series. Non-constant parameters: We are assuming β 1, β 2 stay same throughout sample. If they don t then result may be heteroskedasticity. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

25 Heteroskedasticity: Solving Active: Add the regressors you think cause heteroskedasticity. Passive: Calculate Heteroskedasticity Consistent Standard Errors (HCSE). Heteroskedasticity affects standard errors: Become bigger. These standard errors are robust to heteroskedasticity. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

26 What Distributions are Problematic? Do not assume Normal distribution but Normal distribution is nice distribution. Assume ɛ t iid(0, σ 2 ), so: Distribution unchanging over time. Distribution symmetric around zero. Density Density Density Density Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

27 Normal Distribution: Consequences and Causes Normality not assumed for OLS so provided iid assumption holds inference unaffected: OLS estimator unbiased, consistent, and efficient. But: Non-Normality may reveal information about dataset: Inappropriate functional form and/or data transformation. Informally test by plotting data series: Density Residuals N(s= ) QQ plot Residuals normal Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

28 Testing more formally Skewness and kurtosis of distribution are like mean and variance: They describe shape of distribution: Skewness: How symmetric is the distribution? Kurtosis: How flat/spiky is distribution? Test statistics: χ 2 skewness = T ˆκ2 3 6 χ2 1, (9) χ 2 kurtosis = T ˆκ χ2 1, (10) χ 2 normality = χ2 skewness + χ2 kurtosis χ2 2. (11) PcGive reports this test. Null hypothesis is Normal residuals. Hence if test rejected (we see stars) then our model is misspecified. Test in Baseball model: Normality test: Chi 2 (2) = [0.9400] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

29 χ 2 -Distribution Under null-hypothesis: Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

30 Normal Distribution: Solving Example: Normality test: Chi 2 (2) = [0.9400] We should investigate non-normalities: 1 Non-Normality may be harmless for inference. 2 Non-Normality may reveal important information about model specification. 1 Non-Normality may be harmless for inference: iid assumption translates into non-skewed distribution. Hence if test fail caused by excess kurtosis then model is fine. 2 May reveal information about model to help diagnose other problems found. Inappropriate data transformation. Structural breaks (see recursive testing later) Additional information via Test menu and Test... Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

31 Testing for Functional Form Economic theory often predicts which variables matter. Less often predicts mathematical form of dependence. Test for functional form: RESET test (Ramsey, 1969). 2 Include squares, cubes of fitted values. Null hypothesis of correct functional form: Additional variables do not matter. Auxiliary regression: Y i = β 1 + β 2 X 2,i + β 3 X 3,i + β 4 X 4,i + ψ 1Ŷ2 i + ψ 2Ŷ3 i + v i. Null hypothesis: F-test statistic: H0 : ψ 1 = ψ 2 = 0. (12) F RESET = (RSS R RSS U )/2 RSS U /(T K 2) F 2,T K 2. (13) RESET23 test: F(2,47) = [0.5127] 2 REgression Specification Error Test. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

32 Baseball Attendance: Specification Residuals (scaled) Residuals N(s= ) Density 350 Random vs. Patterns? Normal? AR 1-2 test: F(2,47) = [0.0010]** ARCH 1-1 test: F(1,50) = [0.2773] Normality test: Chiˆ2(2) = [0.9400] Hetero test: F(4,47) = [0.1969] Hetero-X test: F(5,46) = [0.2983] RESET23 test: F(2,47) = [0.5127] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

33 Note: Model Specification - Logs Common Model Specification: Log Model So far: Y t = β 1 + β 2 X t + ɛ t (14) β 2 = dy t dx t Log-log Model: interpret as percentage changes/elasticities log(y) t = β 1 + β 2 log(x t ) + ɛ t (15) β 2 = dlog(y) t dlog(x) t X t dy t Y t dx t X t = dy t dx t X t Y t = elasticity of Y t with respect to Percentage change! For a 1% change in X t expect a β 2 % change in Y t. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

34 Static to Dynamic So far: Test theory using econometric model. E.g. Baseball attendance: positive relationship between winning percentage and attendance. Model: %Attendance t = β 1 + β 2 PCT t + β 3 Unemp. t + ɛ t So far: All variables dated t: This is a static regression. But where variables are today depends on where they were previously. How can we predict variables if all our dates are t? Today: Dynamic models. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

35 Autocorrelation: Correlations through Time May be that variable is correlated with itself at a previous point in time. E.g.: Exchange rate, interest rates, inflation (low and stable prices), unemployment, baseball attendance. Correlation through time is concept called time dependence. We recognise how economic variable today is dependent on where it was yesterday. Expect persistence in many economic variables; especially prices. Supply reflects production capacity which does not drastically change. Demand reflects frequent purchases if a necessity good. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

36 What is a lag? Y t r is lagged variable: The same variable at previous point in time. Y t 1 is first lag: First previous time period. Y t 2 is second lag, Y t 3 third, so on... Use lags to learn about: Persistence: How important is a variable s history? Effects of explanatory variables distributed over time (e.g. advertising). Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

37 E.g. Spot exchange rate: Demand and supply of UK pounds for Japanese Yen (Top), % Attendance (bottom). 250 Yen per British Pound %Attendance Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

38 Autocorrelation Want to understand persistence: Tells us much about economic variables. E.g. price efficiency, partial adjustments, interest rate smoothing. If we don t model it properly, can cause big mistakes. Autoregressive models: Regression model of variable Y t on itself in previous time period Y t 1. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

39 Autoregressive Model Autoregressive model has three elements: (1) Where Y t was the last time period. (2) The unexpected event ɛ t. (3) Constant term allowing mean of Y t to be non-zero. Y t = α }{{} 0 + α 1 Y }{{ t 1 } (3) (1) + ɛ t }{{} (2), ɛ t N[0, σ 2 ]. (16) Notation: normally use α for autoregressive, but equivalent to: Y t = β 1 + β 2 Y t 1 + ɛ t, ɛ t N[0, σ 2 ]. (17) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

40 AR(1) model allows us to determine many things about theory: α 1 : How quickly equilibrium re-established. α 0 and α 1 : Whether equilibrium is zero or otherwise. σ 2 : How much variation there is in Y t around equilibrium. How big are the unexpected events? What is equilibrium value? Again expectations: EY t = α 0 + α 1 EY t 1. (18) Since EY t = EY t 1 we find that µ Y = EY = α 0 /(1 α 1 ). We define µ Y to be the equilibrium value, or unconditional mean of Y t. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

41 Persistence We learn about the persistence of deviations from equilibrium from α 1. To see why note that µ Y = α 0 /(1 α 1 ) implies α 0 = µ Y (1 α 1 ) so that: Y t = α 0 + α 1 Y t 1 + ɛ t = Y t µ Y = α 1 (Y t 1 µ Y ) + ɛ t. (19) We have de-meaned Y t : We only care about α 1 and deviations from equilibrium. If assume no more shocks happen can see how quickly impact of shock disappears. Y t µ Y = α 1 (Y t 1 µ Y ) and Y t 1 µ Y = α 1 (Y t 2 µ Y ) so: Y t µ Y = α 2 1(Y t 2 µ Y ). (20) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

42 We can carry on doing this: Y t µ Y = α 3 1(Y t 3 µ Y )... Y t µ Y = α k 1 (Y t k µ Y ) (21) It so happens that: Corr [Y t, Y t k ] = Cov(Y t, Y t k ) V(Yt ) V(Y t k ) = αk 1 σ2 Y σ Y σ Y = α k 1. (22) Have measured autocorrelation, or correlation through time, of Y t from α 1! The bigger is α 1 and hence nearer to 1, the more persistent is the series: If α 1 = 0.9 then α 2 1 = 0.81 and α 10 1 = If α 1 = 0.2 then α 2 1 = 0.04 and α Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

43 May need more than one lag to explain dynamics of variable: If we model p lags, we have AR(p) model. E.g. AR(2): Y t = α 0 + α 1 Y t 1 + α 2 Y t 2 + ɛ t. Estimators like in multivariate regression: ˆα 2 asks Y t 1 to be still! It controls for first lag to get only second lag effect. ˆα 2 = T t=2 Y t 2(Y t Y t 1 ) T t=2 Y t 2(Y t 2 Y t 1 ). Unconditional mean, variance and covariance affected. E.g. unconditional mean: µ Y = α 0 1 α 1 α 2. (23) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

44 Autocorrelation Plots Common method for learning about autocorrelation is graphically. Autocorrelation function (ACF): Corr [Y t, Y t p ], p = 1, 2,..., 20. Partial ACF (PACF): Corr [Y t, Y t p Y t 1,..., Y t p+1 ], p = 1, 2,..., 20. Number of significant PACF lags = number of lags needed in model. 1 ACF-%Attendance PACF-%Attendance ACF-US_Unemployment Rate PACF-US_Unemployment Rate Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

45 Auto-regressive Model Autoregressive model for Baseball Attendance: %Attendance t = µ 0 + α 1 %Attendance t 1 + ɛ t (24) Estimated Model: %Attendance t = ( ) (0.0935) %Attendance t %Attendance Fitted Residuals (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

46 Autocorrelation Tests and More Lags Earlier: Misspecification testing: Check iid assumption. ˆɛ t all approximately same size? ˆɛ t all unrelated to each other? Latter failure in time series is residual autocorrelation: Corr(ɛ t, ɛ t 1 ) 0. Omitted variable bias: We omit lags that matter for explaining Y t. 2.5 Residuals (scaled) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

47 Independent Distribution: Detection Test using auxiliary regression: ˆɛ t = γ 0 + γ 1 Y t 1 + φ 1ˆɛ t 1 + v t, (25) Null hypothesis: H0 : γ 1 = φ 1 = 0, (26) Absence of autocorrelation in errors of model (model well specified). F-test statistic: F AR = R 2 AR /r ( 1 R 2 AR ) / (T r) F r,t r. (27) Can include up to r lags when testing. Can vary in Test menu and Remedy autocorrelation by including time-dependent information: More lags or extra variables (with lags). Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

48 Last Test: ARCH ARCH: Autoregressive Conditional Heteroskedasticity. Heteroskedasticity: ɛ t (0, σ 2 t ), variance changes over time. Autoregressive: Variable correlated with itself. ARCH: σ 2 t = α 0 + α 1 ɛ 2 t 1 + u t. Variance is persistent: Spells of high and low volatility. Common in financial markets. Impact on regression model same as heteroskedasiticity: Efficiency. Test using AR(1) on squared residuals: ˆɛ 2 t = α 0 + α 1ˆɛ 2 t α rˆɛ 2 t r + v t. F-test: H0 : α 1 = = α r = 0. Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

49 Back to Baseball... Recall our model: Model: %Attendance t = β 1 + β 2 PCT t + β 3 Unemp. t + ɛ t Misspecification Tests AR 1-2 test: F(2,47) = [0.0010]** ARCH 1-1 test: F(1,50) = [0.2773] Normality test: Chiˆ2(2) = [0.9400] Hetero test: F(4,47) = [0.1969] Hetero-X test: F(5,46) = [0.2983] RESET23 test: F(2,47) = [0.5127] Residual Autocorrelation! Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

50 Baseball Attendance: Specification Residuals (scaled) Density Residuals N(s= ) 350 Random vs. Patterns? Normal? AR 1-2 test: F(2,47) = [0.0010]** ARCH 1-1 test: F(1,50) = [0.2773] Normality test: Chiˆ2(2) = [0.9400] Hetero test: F(4,47) = [0.1969] Hetero-X test: F(5,46) = [0.2983] RESET23 test: F(2,47) = [0.5127] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

51 Add extra lags of dependent variable: %Attend. t = α 1 %Attend. t 1 + β 1 + β 2 PCT t + β 3 Unemp. t + ɛ t %Attend. = (0.0913) ( ) %Attend. t ( ) Unempl. t ( ) PCT t AR 1-2 test: F(2,45) = [0.5510] ARCH 1-1 test: F(1,49) = [0.4158] Normality test: Chiˆ2(2) = [0.0036]** Hetero test: F(6,44) = [0.7810] Hetero-X test: F(9,41) = [0.9109] RESET23 test: F(2,45) = [0.8994] What now?? Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

52 Look at the model fit and residuals: %Attendance Fitted Residuals (scaled) Large Residual! Large Residual! May distort normality tests! What happened? More on Thursday! Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

53 Practical Computer Lab Session 2: Testing joint-hypotheses F-test Interpreting Misspecification Tests Investigating Time Series Properties Time Series plots Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

54 Practical Fulton Fish Market: Price, Quantity, Weather Load fish.in7 Series Model for qty = log(quantity) Weather: Stormy, Rainy, Cold Graph the series! (Important first step!) Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

55 Autoregressive Models Construct Auto-regressive models for log(quantity) sold: Determine lag length: (Partial) Auto-correlation function (max 10 lags) Estimate an AR(1), AR(2) models What is the long-run equilibrium? Interpret mis-specification tests Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

56 AR(1) Model Output EQ(15) Modelling qty by OLS The estimation sample is: Coefficient Std.Error t-value t-prob Part.Rˆ2 qty_ Constant sigma RSS Rˆ F(1,108) = [0.033]* Adj.Rˆ log-likelihood no. of observations 110 no. of parameters 2 mean(qty) se(qty) AR 1-2 test: F(2,106) = [0.1422] ARCH 1-1 test: F(1,108) = [0.1514] Normality test: Chiˆ2(2) = [0.0316]* Hetero test: F(2,107) = [0.0282]* Hetero-X test: F(2,107) = [0.0282]* RESET23 test: F(2,106) = [0.4989] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

57 Expand the model Interested in effects of weather on quantity sold: Estimate auto-regressive model with weather variables added in Include: Stormy, Rainy, Cold Which variables are individually significant? Which variables are jointly significant? Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

58 Model Output: Weather Effects EQ(17) Modelling qty by OLS Coefficient Std.Error t-value t-prob Part.Rˆ2 qty_ Constant stormy rainy cold sigma RSS Rˆ F(4,105) = [0.036]* Adj.Rˆ log-likelihood no. of observations 110 no. of parameters 5 mean(qty) se(qty) AR 1-2 test: F(2,103) = [0.4410] ARCH 1-1 test: F(1,108) = [0.1845] Normality test: Chiˆ2(2) = [0.0128]* Hetero test: F(5,104) = [0.2352] Hetero-X test: F(5,104) = [0.2352] RESET23 test: F(2,103) = [0.5198] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

59 Test for Joint Significance F-Test for joint significance of all 3 weather variables Test - Exclusion Restrictions Select all 3 variables Test for excluding: [0] = stormy [1] = rainy [2] = cold Subset F(3,105) = [0.1233] Felix Pretis (Oxford) Time Series Akita Intl. University, / 60

60 Further Exercises Modelling share price (Apple) using Google search data: Can share price be modelled using consumer interest (measured through google search term: iphone ) load data apple google.in7 Build a model for Apple closing price What lag length should you choose? Is the google search term a useful predictor? What about lagged search terms? What about the effect of google searches on the trading volume? Felix Pretis (Oxford) Time Series Akita Intl. University, / 60